# Is the cap product the same on cochain complexes with isomorphic cohomology rings?

Let $$X$$ be a topological space. Let $$\mathcal{D}$$ be a cochain algebra, with cohomology ring $$H^* (\mathcal{D})$$. Suppose that we have that the cohomology ring $$H^* (X)$$ (endowed with the cup product) is isomorphic, as a ring, to $$H^* (\mathcal{D})$$, equipped with its respective operation (and the homology groups of $$X$$ are isomorphic to the homology groups of $$\mathcal{D}$$).

There is an action of $$H^* (X)$$ on $$H_* (X)$$ given by the cap product: $$\frown:H_* (X) \otimes H^* (X) \rightarrow H_* (X)$$. There is also an action of $$H^* (\mathcal{D})$$ on $$H_* (\mathcal{D})$$ given by dualizing the algebra structure of $$\mathcal{D}$$ (described below). I am interested in whether this "cap product" on $$\mathcal{D}$$ induces the same action of $$H^* (\mathcal{D})$$ on $$H_* (\mathcal{D})$$ as the cap product does for the cohomology and homology of $$X$$.

Denote by $$*$$ the multiplication in $$\mathcal{D}$$. Let $$\widehat{\mathcal{D}}$$ denote the chain complex obtained by dualizing the cochain complex $$\mathcal{D}$$. We obtain an operation

$$\frown:\mathcal{D} \times \widehat{\mathcal{D}} \rightarrow \widehat{\mathcal{D}}$$

by defining $$\langle u * v, w \rangle = \langle u, v \frown w \rangle$$, the dual of $$*$$.

This is the exact way in which, if we were given the cup product on the level of cochains of $$X$$, we would dualize to get the cap product on the cochains/chains of $$X$$.

There is an induced operation on homology:

$$\frown:H_* (\mathcal{D}) \otimes H^* (\mathcal{D}) \rightarrow H_* (\mathcal{D}).$$

Is this operation the same (isomorphic to) the cap product on the isomorphic cohomology and homology groups of $$X$$? I think that the answer is yes, but I cannot find a proof.

EDIT: Additionally, I would at least like to know if I am correct in thinking that this is always true in the case of zero torsion.

• Do you have non-trivial examples where this is true? I would honestly be surprised if it worked in general. Jul 2, 2020 at 13:58
• Is this not always true if I have zero torsion? So that cup product on the level of cohomology completely determines cap?
– Matt
Jul 4, 2020 at 20:22